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Medicine show: A Calogero model with principal series states

by Tarek Anous, Jackson R. Fliss, Jeremy van der Heijden

Submission summary

Authors (as registered SciPost users):

Tarek Anous

Abstract

The{\it Calogero model} is an interacting, $N$-particle, $\mathfrak{sl}(2,\mathbb{R})$-invariant quantum mechanics, whose Hilbert space is furnished by a tower of discrete series modules. The system enjoys both classical and quantum integrability at any $N$ and at any value of the coupling; this is guaranteed by the existence of $N$ mutually-commuting currents, one of them being the Hamiltonian. In this paper, we alter the Calogero model so that it may accommodate states in the unitary principal series irreducible representation of $\mathfrak{sl}(2,\mathbb{R})$. Doing so requires changing the domain of the quantum operators---a procedure which succeeds in preserving unitarity and $\mathfrak{sl}(2,\mathbb{R})$-invariance, but breaks the integrability of the theory. We explicitly solve the deformed model for $N=2,3$ and outline a procedure for solving the model at general $N$. We expect this deformed model to provide us with general lessons that carry over to other systems with states in the principal series, for example, interacting massive quantum field theories on de Sitter space.

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